What are the ingredients for an effective mathematics lesson? Teachers are continually faced with a range of advice or ideas to improve their mathematics lessons and often this just creates confusion. It’s a little bit like being a cook. New recipes appear online and in cookbooks on bookstore shelves, but often they’re just adaptations of classic recipes that have been around before, their foundation ingredients are tried and tested, and often evidence based. There are always the staple ingredients and methods that are required for the meal to be successful.

The following is a list of what I consider to be important ingredients when planning and teaching an effective mathematics lesson. The list (or recipe) is split into two parts: lesson planning and lesson structure.

Lesson planning:

Be clear about your goal. What exactly do you want your students to learn in this lesson? How are you going to integrate mathematical content with mathematical processes? (The proficiencies or Working Mathematically components) Will you consider the General Capabilities in your planning?

Know the mathematics. If you don’t have a deep understanding of the mathematics or how students learn that aspect of mathematics, how can you teach it effectively? Where does the mathematics link across the various strands within the mathematics curriculum?

Choose good resources. Whether they are digital or concrete materials, make sure they are the right ones for the job. Are they going to enhance students’ learning, or will they cause confusion? Be very critical about the resources you use, and don’t use them just because you have them available to you!

Select appropriate and purposeful tasks. Is it better to have one or two rich tasks or problems, or pages of worksheets that involve lots of repetition? Hopefully you’ve selected the first option – it is better to have fewer, high quality tasks rather than the traditional worksheet or text book page. You also need to select tasks that are going to promote lots of thinking and discussion.

Less is more. We often overestimate what students will be able to do in the length one lesson. We need to make sure students have time to think, so don’t cram in too many activities.

You don’t have to start and finish a task in one lesson. Don’t feel that every lesson needs to be self-contained. Children (and adults) often need time to work on complex problems and tasks – asking students to begin and end a task within a short period of time often doesn’t give them time to become deeply engaged in the mathematics. Mathematics is not a race!

Lesson Structure:

Begin with a hook. How are you going to engage your students to ensure their brains are switched on and ready to think mathematically from the start of each lesson? There are lots of ways to get students hooked into the lesson, and it’s a good idea to change the type of hook you use to avoid boredom. Things like mathematically interesting photographs, YouTube clips, problems, newspaper articles or even a strategy such as number busting are all good strategies.

Introduction: Make links to prior learning. Ensure you make some links to mathematics content or processes from prior learning – this will make the lesson more meaningful for students and will reassure anxious students. Use this time to find out what students recall about the particular topic – avoid being the focus of attention and share the lesson with students. Talk about why the topic of the lesson is important – where else does it link within the curriculum, and beyond, into real life?

Make your intentions clear. Let students know what they’re doing why they’re doing it. How and where is knowing this mathematics going to help them?

Body: This is a good time for some collaboration, problem solving and mathematical investigation. It’s a time to get students to apply what they know, and make links to prior learning and across the mathematics curriculum. This is also a time to be providing differentiation to ensure all student needs are addressed.

Closure: This is probably the most important time in any mathematics lesson. You must always include reflection. This provides an opportunity for students to think deeply about what they have learned, to make connections, and to pose questions. It’s also a powerful way for you, the teacher, to collect important evidence of learning. Reflection can be individual, in groups, and can be oral or written. It doesn’t matter, as long as it happens every single lesson.

There are many variables to the ingredients for a good mathematics lesson, but most importantly, know what and how you are teaching, provide opportunities for all students to achieve success, and be enthusiastic and passionate about mathematics!

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies: Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities, one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson. Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys are also very worthwhile tasks. For good mathematical problems go to the nrich website. For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300 (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

higher-level thinking within authentic and meaningful contexts;

complex problem solving;

open-ended responses; and

substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

In one of my previous posts I wrote about the use of children’s literature to encourage rich mathematical investigations and improve student engagement with mathematics. One of my favourite books, Math Curse by John Szieska and Lane Smith, is described in the blog post as a great way to engage reluctant learners. Even better, Math Curse encourages children (and their teachers) to see the mathematics that is embedded in every aspect of our lives. In this post I am going to share some student work from a Grade 3 classroom. In this classroom, the teacher read the book to the students before challenging them create their own class maths curse. The children took their own photographs, and working in small groups, they came up with a range of mathematical problems and investigations, which they then gave to other groups to solve.

Here are some of the photos with their accompanying questions:

Beyblades:

If one of the beyblades spins for 2 minutes and 31 seconds and the other one spins for 1 minute and 39 seconds what is the difference between the two times?

If one of the beyblades spins for 1 minute and 1 second and another spins for 78 seconds, which beyblade spun for the longest and by how long?

Hair:

If there are 31 people in the class (10 boys and 21 girls) and all of them have hair that is 30cm long. Half of the boys cut 10cm off their hair, the other half cut 20cm off their hair. How long is the classes hair now altogether? How long was it before? How much hair has been cut altogether?

Check your friend’s hair. Estimate how long it is when it is out, how long it is when it is in a ponytail, and how long it is when it is in a braid. List some different ways you could check if your estimate is accurate? What are the potential problems with your methods?

I’m 9 years old. I had really long hair for 6 years, then I cut it. How long did I have short hair for?

I have 5 friends that are girls and 2 friends that are boys. All 5 girls have hair length of 50cm. The boys both have different lengths of hair. The 1st boy has 30cm of hair, the second has 25cm of hair. What is the difference between the 1st boy and the girls and the 2nd boy and the girls?

Birthday Balloons:

Write down the dates of important celebrations. If you add all the dates together, what is the value of their numbers?

How many days are there in 6 years?

If everyone’s birthday occurred every three years (starting the year you are born) what years would your birthday fall on?

If Lisa and Jane went on a holiday every 2 months, how many holidays could they take in a year?

If you could rearrange the seasons, what months would you choose to be Spring? Why?

What is the most popular letter in the days of the months?

Why do you think there are 4 seasons in a year?

From Problem Solving to Problem Posing

What is the purpose of getting students to write mathematical problems? First of all, the problems give us good insight into whether students recognise mathematical situations, and whether they understand where, how, and what mathematics is applied in day to day situations. An added bonus is that the students are highly engaged because they have ownership of the mathematics they are generating, the topics they choose are of interest to them, and stereotypical perceptions of school mathematics are disrupted.

Student Reflection

The students who wrote the examples above completed a structured written reflection following the sequence of designing and solving each others’ maths curses. Here are some of reflection prompts and a sample of responses:

What did you enjoy about today’s learning?

“working with my team” “working at the problems for a long time and then finally getting them after a long, hard discussion”

“I liked the chess card the best because we had to solve it together and use problem solving”

“having a go at tricky questions even if i got them wrong”

Did you learn anything new?

“how to work things out in different ways”

“working in groups helps you learn more skills”

“not every question uses just one skill like addition, division, multiplication or subtraction”

“when I am challenged I learn more”

“Maths is not always easy”

“how to work together”

“Everyone in the group has different responses so we needed proof to figure out the right one”

What surprised you about this task?

“It surprised me how hard my own questions were”

“I didn’t know that we could come up with so many interesting questions” “I got a shock! We had to research to solve some problems, Adam even taught me how to add a different way”

“I got some questions wrong “

“It was hard but if we put our brains into gear we could figure it out”

“I was able to play while doing maths”

Using activities such as this provides multiple benefits for students. Contextualising the mathematics using students’ interests highlights the relevance of the curriculum, improves student engagement, and makes mathematics meaningful, fun and engaging!

Many children begin the new school year with feelings of fear and anxiety. Will they like their new teacher or teachers? Will the work be difficult? What will the homework be like? As you prepare programming and planning for a new teaching year and new students, give some thought to the strategies and activities you and your students can do in the first few weeks of term to ensure everyone gets the most out of their mathematics lessons for the entire school year. Think about what you can do differently in 2017 to make your work more engaging for both you and your students. The following are some ideas to consider.

Be a positive mathematical role model

I’m sure this won’t come as a surprise, but there are teachers in our schools who actually don’t like maths and don’t like teaching it. Why is this a problem? Student know! This knowledge perpetuates the common misconception that it’s okay to dislike mathematics, and worse still, it’s okay to be considered ‘bad’ at maths. Unless the teacher is an award-winning actor or actress, it’s really difficult to hide how you feel about a subject – it’s obvious in body language, tone of voice and of course, the way you teach the subject and the resources you use. If you know someone like this, suggest they seek some support from a colleague or colleagues. Often the reason a person dislikes mathematics is related to a lack of confidence.

Get to know your students as learners of mathematics

The foundation of student engagement requires an understanding of students as learners, in other words, the development of positive pedagogical relationships (Attard, 2014). Positive relationships require teachers to understand how their students learn, and where and when they need assistance. It’s also important to provide opportunities for ongoing interactions between you and your students as well as amongst your students.

Another way to get to know your students as learners is to use existing data. For example, if your school takes part in external testing such as PAT, you can use this data as a guide. However, keep in mind that things change quickly when children are young – what they knew or understood three months ago may be very different after a long summer holiday.

A great activity to do in the very first few maths classes of the year is to ask your students to write or create a ‘Maths Autobiography’. If required, provide the students with some sentence starters such as “I think maths is…” “The thing I like best about maths is…” “The thing or things that worry me about maths is…” They could do this in different formats:

In a maths journal

Making a video

Using drawings (great for young children – a drawing can provide lots of information)

Start off on a positive note

Have some fun with your maths lessons. I would strongly recommend that you don’t start the year with a maths test! If you want to do some early assessment, consider using open-ended tasks or some rich mathematical investigations. Often these types of assessments will provide much deeper insights into the abilities of your students. You can even use some maths games (either concrete or digital) to assess the abilities of your students.

A great maths activity for the first lesson of the year is getting-to-know-you-mathematically, where students use a pattern block and then need to go on a hunt to find other students who have specific mathematical attributes. Encourage your students to find someone different for every attribute on the list, and change the list to suit the age and ability of your students. For example, in the younger years you could use illustrations and not words. In the older years, you could make the mathematics more abstract.

Take a fresh look at the curriculum

Even if you’ve been teaching for many years, it’s always good to take a fresh new look at the curriculum at the start of each year. Consider how the Proficiencies or Working Mathematically processes can be the foundation of the content that you’re teaching. For example, how can you make problem solving a central part of your lessons?
Take a close look at the General Capabilities. They provide a perfect foundation for contextual, relevant tasks that allow you to teach mathematics and integrate with other content areas.

Consider the resources you use: Get rid of the worksheets!

Think about using a range of resources in your mathematics teaching. Regardless of their age or ability, children benefit from using concrete manipulatives. Have materials available for students to use when and if they need them. This includes calculators in early primary classrooms, where students can explore patterns in numbers, place value and lots of other powerful concepts using calculators.

Children’s literature is also a great resource. A wonderful book to start off the year is Math Curse by Jon Scieska and Lane Smith. Read the book to your students either in one sitting or bit by bit. There are lots of lesson ideas within the pages. Ask your students to write their own maths curse. It’s a great way to illustrate that mathematics underpins everything we do! It’s also a great way to gain insight into how your students view mathematics and what they understand about mathematics.

How will you use technology in the classroom?

If you don’t already integrate technology into your mathematics lessons, then it’s time to start. Not only is it a curriculum requirement, it is part of students’ everyday lives – we need to make efforts to link students’ lives to what happens in the classroom and one way to do that is by using technology. Whether it’s websites, apps, YouTube videos, screencasting, just make sure that you have a clear purpose for using the technology. What mathematics will your students be learning or practicing, and how will you assess their learning?

Reach out to parents

As challenging as it may be, it’s vital that parents play an active role in your students’ mathematical education. They too may suffer from anxiety around mathematics so it’s helpful to invite them into the classroom or hold mathematics workshops where parents can experience contemporary teaching practices that their students are experiencing at school. Most importantly, you need to communicate to parents that they must try really hard to be positive about mathematics!

These are just a few tips to begin the year with…my next blog post will discuss lesson structure. In the meantime, enjoy the beginning of the school year and:

Be engaged in your teaching.

Engaged teachers = engaged students.

Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34(3), 1-14.

It’s that time of year again! Last year I wrote a series of blog posts to encourage teachers to continue to provide rich teaching and learning activities until the very end of the school year. I thought it would be a good idea to repost these activities for those who might want a reminder of some engaging Christmas themed mathematical explorations.

As the end of the school year approaches, report writing is almost complete and Christmas is on the horizon. It’s easy to lose focus, get distracted, and keep students occupied with ‘busy work’. However, it’s critical that we don’t waste a minute of students’ learning time, particularly when we know that over the long Christmas break some students may regress in relation to mathematical fluency and understanding.

So how can you keep mathematics engaging until the last day of school? Consider the elements required for sustained engagement to occur. Three factors are critical: cognitive, operative, and affective engagement. In terms of mathematics, true, sustained engagement occurs when students are procedurally engaged and interacting with the mathematics and with each other; when there is an element of cognitive challenge within the task; and when they understand that learning mathematics is worthwhile, valuable, and useful both within and beyond the classroom. It is easy to mistakenly think that students are engaged when they appear to be busy working, or ‘on task’. True engagement is much deeper than ‘on task’ behaviour, rather, it can be viewed as ‘in task’ behaviour, where all three elements; cognitive, operative and affective, come together. This leads to students valuing and enjoying school mathematics and seeing connections between the maths they do at school and the maths they use in their lives outside school (for more information see my FRAMEWORK FOR ENGAGEMENT WITH MATHEMATICS).

As this time of year it is easy to design mathematics tasks that promote high engagement and have the potential to stimulate learning. The following is a set of problem solving activities based on the famous Christmas Carol, The Twelve Days of Christmas. The activities are suitable for children in the middle years (grades 5 to 8), however can be easily adapted to suit younger or older learners.

Resources Required:

A copy of the lyrics of “The Twelve Days of Christmas”

The book “The Twelve Days of Christmas” (There are several versions available)

Price Lists

Other resources as required, eg. shopping catalogues

Possible Investigations starters/Task cards

Teaching/Learning Activities:

Read the book/lyrics or listen to the song “The Twelve Days of Christmas”

Discuss how the gift giver has to increase the number of gifts to his true love each day.

Provide students with a price list (this can be adapted according to the ability of the group)

At this point students can be asked to investigate the cost of the gifts, turning the activity into an open-ended investigation, or, specific questions can be posed to the students.

Examples of possible problems to explore are:

What is the total number of gifts given?

Is there an easy way to work this out?

What is the total cost of the gifts?

If the department store was holding a pre-Christmas sale and offered a 15% discount for all purchases, what would the new cost be? (This does not include performers, maids, lords etc)

What if the discount offered only applied to live animals?

The maids give a 10% when booked for more than two consecutive days. What would their new fee be?

The musicians charge a 10% Goods and Service tax and this must be added to the total cost.

How many people arrive at the true love’s house on the twelfth day?

What would it cost to feed all the people and the animals? (Internet would come in handy here!)

Use some Christmas shopping catalogues to replace the gifts with something more appropriate for a modern true love and calculate the cost.

Find a range Christmas Advent calendars at Nrich where each day in December has new mathematical problems and investigations to explore http://nrich.maths.org/7778

All of the above activities have the potential to promote high levels of engagement at a time of year when it is difficult for students and teachers to remain focussed. However, it is important to remember that any activity is only as good as the teacher implementing it. To enhance students’ engagement and learning, ensure there is regular student reflection, and rich discussion about the mathematics embedded with the tasks.

This post was originally published last year, and I thought it timely to republish considering many teachers in Australia are busy spending the school holidays programming and planning for Term 2.Often when I work with teachers I am asked for advice regarding the design of a scope and sequence for mathematics. The programming and planning of mathematics seems to cause much concern, and often the reason is that there is no ‘magic fix’ or one-size-fits-all solution.

Traditionally, schools have planned their mathematics teaching using a topic-by-topic or strand-by-strand approach. Sometimes there is a formula for teaching the Number and Algebra strand for a certain number of days per week, with the other days dedicated to the remaining syllabus strands. Often, the strands are split into single, stand-alone topics. Unfortunately, there are issues with this approach. Teaching individual topics in mathematics hinders students in gaining a deep understanding of mathematics and the connections that exist between and among the strands. Teaching in this way can promote a traditional, rote learning approach where the opportunities for mathematical thinking are limited. Our curriculum places the Proficiencies (Working Mathematically in New South Wales) at the forefront of teaching and learning mathematics – teaching topics in isolation does not promote the Proficiencies.

So what’s the solution? Consider planning and programming using a ‘big idea’ approach. What’s a big idea? Big ideas are hard to define and different people have differing ideas on what the big ideas in mathematics actually are. However, all the definitions in literature have one thing in common – they all refer to big ideas as the key to making connections between mathematical content and mathematical actions, and they all link mathematical concepts. Take, for example, the big idea of equivalence. This relates to number and numeration, measurement, number theory and fractions, and algebraic expressions and equations. Connections can be made across the strands and these links should be made explicit to students.

Charles (2005) presents a total of 21 big ideas across the mathematics curriculum, however he states that these are not fixed – they can be adapted. He also states that a big ideas approach has implications for curriculum and assessment and professional development – teachers need to develop their pedagogical content knowledge to ensure they have a deep understanding of the connections within the curriculum if they are to teach mathematics successfully.

Of course, there are challenges to teaching using a big ideas approach. Teachers often feel under pressure to address all curriculum outcomes, and often this is the reason that the topic-by-topic approach is adopted. Using a big ideas approach can feel messy – it is not linear and in some ways feels as though it is conflicting with the organisation of our curriculum. However, we must remember that although our curriculum is separated into strands and sub-strands, this is simply an organisational tool and does not mean that mathematics should be taught in this same way.

My advice would be to take our curriculum, pull it apart and try seeing it differently – what areas of the curriculum have obvious links? How can you link aspects of measurement to the number strand? Where does measurement and geometry link? And how can you use the statistic and probability strand to teach number concepts? Making connections will make your teaching easier in the long run, and more importantly, will result in deeper learning and deeper engagement with mathematics.

Randall, C. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. NCSM Journal, 7(3), 9-24.